# Mathematical Diversions

A collection of mathematical miscellany that diverts me from my main research interests. Some of the posts are interesting (to me) mathematical problems with their solutions. Others are mostly recreational things that distracted me at one time or another. The intended audience for posts in this category are my sole self, and so I make no great effort to polish the writing. Do send me an email with questions if you find anything here interesting or confusing.

### Lengths of Parallel Curves

Let's start with something simple: Given a circle with radius $R$, draw a concentric circle with radius 1 larger. How much longer is …

### A Tangent Identity

Here's a slightly non-obvious relation between some angles, that I ran into accidentally. Claim: (angles below given in degrees) \[ …

### An Implicit Assumption about Expectation Values

A short note about an implicit assumption behind the notion of expectation values, and how it relates to economics applications.

### Oscillatory Solutions to the Heat Equation

As we all learned in a first course in PDEs, for "reasonable" initial data $u_0$, the solution to the linear heat equation …

### Some Periodic Functions

Some non-intuitive facts about periodic functions

### Traceless general relativity

A quick summary of the traceless version of general relativity.

### The Nirenberg Trick and Wave Maps

The "Nirenberg trick" or "Nirenberg transformation" is the observation1 that the semilinear wave equation …

### Convergence to Planewaves for Solutions to Schrödinger's equation III

In this installment, we look at Schrödinger's equation with barriers, and study the decay of the solutions.

### Convergence to Planewaves for Solutions to Schrödinger's equation II

We use Laplace transform arguments to look at the convergence of infinite energy solutions of the free 1D Schrödinger's equation …

### Convergence to Planewaves for Solutions to Schrödinger's equation

We look at the convergence of infinite energy solutions of the free 1D Schrödinger's equation to plane waves.